Orderings, Valuations and Quadratic Forms

Orderings, Valuations and Quadratic Forms

by T. Y. Lam, Tsit-Yuen Lam
     
 

ISBN-10: 0821807021

ISBN-13: 9780821807026

Pub. Date: 12/31/1983

Publisher: American Mathematical Society

The remarkable relationships and interplay between orderings, valuations and quadratic forms have been the object of intensive and fruitful study in recent mathematical literature. In this book, the author, a Steele Prize winner in 1982, provides an authoritative and beautifully written account of recent developments in the theory of the ''reduced'' Witt ring of a

Overview

The remarkable relationships and interplay between orderings, valuations and quadratic forms have been the object of intensive and fruitful study in recent mathematical literature. In this book, the author, a Steele Prize winner in 1982, provides an authoritative and beautifully written account of recent developments in the theory of the ''reduced'' Witt ring of a formally real field. This area of mathematics is growing rapidly and promises to become of increasing importance in reality questions in algebraic geometry. The book covers many results from original research papers published in the last fifteen years. The presentation in these notes is largely self-contained; the only prerequisite might be a good working knowledge of general valuation theory and some familiarity with the basic notions and terminology of quadratic form theory. The first chapters of the author's previous book, published by W. A. Benjamin, are a good source for such background material. However, this volume may be read as an independent introduction to ordered fields and reduced quadratic forms using valuation-theoretic techniques. Orderings and valuations are related through the notion of compatibility; valuations and quadratic forms are related through the notion of residue forms, while quadratic forms and orderings are related through the notion of signatures. After a beginning chapter on the reduced theory of quadratic forms, the author lays the foundation for the study of compatibility. This is followed by an introduction to the techniques of residue forms and the relevant Springer theory. The author then presents the solution of the Representation Problem due to Bechker and Brocker, with simplifications due to Marshall. The notion of fans plays an all-important role in this approach. Further chapters threat the theory of real places and the real holomorphy ring, prove Brocker's theorem on the trivialization of fans, and study in detail two important invariants of a preordering (the chain length and the stability index). Other topics treated include the notion of semiorderings, its applications to SAP fields and SAP preorderings, and the valuation-theoretic Local-Global Principle for reduced quadratic forms.

Product Details

ISBN-13:
9780821807026
Publisher:
American Mathematical Society
Publication date:
12/31/1983
Series:
Cbms Regional Conference Series in Mathematics, #52
Pages:
143

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